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Studying again differential geometry, anomalies and topology, I wondered if there is ANY physical interpretations (in terms of QFT or even classical field theory) of the Chern-Simons forms, via vacuum, physical processes or alike. That is, if

$$\omega_{2n+1}=n\int_0^1 dt t^{n-1}Tr[A(dA+tA^2)^{n-1}]$$

Also related, the shift of the 1-form $A\rightarrow A+v$ seems to have a relation with $2n-2$ forms and Faddeev-Popov ghosts fields, but as far as I know, CS forms can only exists in odd dimensions and, together YM, are the only gauge invariant actions. Is that true?

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